Forces & Trigonometry for Physicists

SCALAR: quantities that have magnitude only e.g. Energy, Temperature, Speed

VECTORS : quantities that have a magnitude and direction e.g. Forces, Velocity.

This section involves forces – but the rules below work with any vector.

RULE 1: Formal definition: Two vectors which are the same magnitude, same direction are considered as the same. Translation: We can slide vectors around to make a “chain” or consider them from a point (Whichever makes our lives easier!!). e.g. An object having a mass of 1 kg rests on a very slippery horizontal surface. Two cords are attached to it and these pass over pulleys and have weights at their ends such that one cord pulls on the object with a 10N force and the other with a 5.0N force. The angle between the cords is 40o and the object starts off an equal distance from the two pulleys.

Note: The length of the lines drawn is proportional to the magnitude of the forces. The direction of the lines drawn is the same as the direction of the forces.

Why do we want to slide around the forces? Wouldn’t it be nice if we could combine all these forces to make just one!

Method 1: Putting in a chain.

By putting all the forces in a chain (each force drawn to scale), we can find the “resultant” (what one force is the equivalent of all the forces acting) by drawing a line starting at the beginning of the first force and finishing at the end of the last force. The forces in the chain can be in any order – it will not matter! e.g. Below is an object with a number of forces acting on it.

i.e. All the forces on the object can be brought together to be the equivalent of one force in one direction!!

WorkFGDT  is the same as

If the chain of forces ends up in a ring, then there is no resultant, no overall force and the object is in equilibrium.

However, this relies on accurate drawing – and the resultant may be a decimal!

Method 2: Resolving Forces

If we split each force into 2 forces at right angles to each other – for instance, horizontal and vertical components, we can just add up all the horizontal bits and all the vertical bits. Our chain will boil down to two forces at right angle with each other. Those two with the resultant will make a right-angled triangle. Using Trigonometry and Pythagoras, we can work out the angle that is made by the resultant from, say, the horizontal and its magnitude (length of the vector!).

Part 1 – Splitting up the forces

Here’s the initial problem – what’s the resultant force and the angle which it is acting?

Let’s look at one force at a time ……

The 12N can be split up by trigonometry. (Use the notes from Trig – intro to help)

o H1 = 12 cos 30 = 10.39N

o V1 = 12 sin 30 = 6N

The 15N can also be split up by trigonometry

o H2 = 15 cos 20 = 14.10N

WorkFGDT  o V2 = 15 sin 20 = 5.13N

Part 2 – Putting together the “horizontal” bits and the “vertical” bits.

Now H1 and H2 are in the same direction – so a chain would be just adding them …. H1 + H2 = 24.49N

V1 and V2 are going in the opposite direction to each other – so one will partially cancel out the other. If we count upwards as positive (remember – direction matters with vectors!!)

V1 + V2 = 6 + (- 5.13) = 0.87N Part 3 – Making the resultant

Using Pythagoras

Resultant = 24.492  0.872 = 24.51N

Using Trigonometry

1 0.87    tan   = 2.03o above the  24.49  horizontal

The same principle applies regardless of how many forces or whether it is done horizontally and vertically or parallel and perpendicular to a slope! If you change the directions, just think of tilting everything a bit.

WorkFGDT 